Homework 1b

This homework exercises concepts related to convolutions.

I. A simple proof

Prove that convolution is associative. It is sufficient to do so for discrete-time signals and systems. \[(x[n] \otimes h_1[n]) \otimes h_2[n] = x[n] \otimes (h_1[n] \otimes h_2[n])\]

II. Work out the following convolutions:

Work out the convolutions below. Graph the solutions. For the continuous-time problem, assume $\alpha$ is negative. For the discrete-time problems, $0 \le \alpha < 1.0$
  1. $x[n] \otimes h[n]$ where $x[n] = u[n]$ and the system $h[n] = \alpha^nu[n]$.
  2. $x(t) \otimes h(t)$ where $x(t) = u(t)$ and the system $h(t) = e^{\alpha t}u(t)$. Note that the system here is continuous time.
  3. $x[n] \otimes h[n]$, where $x[n]$ is a unit step, i.e. $x[n] = u[n]$ and system response $h[n]$ is also a unit step $h[n] = u[n]$.
  4. $x[n] \otimes h[n]$, where $x[n]$ is a pulse of width $n_0$ starting at $n=0$ and the system response $h[n]$ is a pulse of width $n_1$ starting at $n=0$. Hint: express $x[n]$ and $h[n]$ in terms of unit steps and draw upon the properties of convolutions in linear shift-invariant systems. You can build up on the solution to part 3.

Due date: Monday, 23th Feb